PrepRung
Sign inSign up free

NDA 2020 Maths · Mathematics · Continuity and Differentiability · hard

Let f(x)=|x|^3. 1. f is twice differentiable at x=0. 2. f'(0)=0 and f''(0)=0.

Single correct option

Let f(x)=∣x∣3f(x)=|x|^3. Consider the following statements:
1. ff is twice differentiable at x=0x=0.
2. f′(0)=0f'(0)=0 and f′′(0)=0f''(0)=0.
Which of the statements is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (C) Both 1 and 2

For x≥0x\ge0, f=x3f=x^3 and for x<0x<0, f=−x3f=-x^3. So f′(x)=3x∣x∣f'(x)=3x|x|, which is continuous with f′(0)=0f'(0)=0. Then f′′(x)=6∣x∣f''(x)=6|x| and f′′(0)=lim⁡h→03h∣h∣−0h=lim⁡3∣h∣=0f''(0)=\displaystyle\lim_{h\to0}\dfrac{3h|h|-0}{h}=\lim 3|h|=0. Thus f′′(0)f''(0) exists and is 00; both statements are correct. The function fails to have a third derivative at 00 because 6∣x∣6|x| has a corner, a fact many candidates over-apply to the lower derivatives.
Report a mistake

Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake